We next present some graph notations.
The closure operation introduced by Bondy and Chvátal [1]}
is a powerful tool for the problems of Hamiltonicity of graphs.
Let \(G\)  be a graph of order \(n\) . The \(s\) -closure of \(G\) ,
denoted by \(\mathrm {cl}_s(G)\) ,
is the graph obtained from \(G\)  by recursively joining pairs of
non-adjacent vertices whose degree sum is at least \(s\)  until no such pair
remains.
It is not hard to prove that the \(s\) -closure
of \(G\)  is uniquely determined; see, e.g., [1]}.
Clearly, \(G\)  is a subgraph of \(\mathrm {cl}_s(G)\)  for every \(s\) ,
and for any two non-adjacent vertices in \(\mathrm {cl}_s(G)\) ,
the sum of their degrees is less than \(s\) .
